Growth functions for some uniformly amenable groups

Dronka, Janusz; Wajnryb, Bronislaw; Witowicz, Paweł; Orzechowski, Kamil

doi:10.1515/math-2017-0049

Cited by 2 publications

(2 citation statements)

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“…The integers Z are an example of a uniformly amenable group. More generally, all solvable groups are uniformly amenable [31,16]. Note that the class of solvable groups includes examples of uncountable groups.…”

Section: Lemma B2 For Everymentioning

confidence: 99%

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An uncountable ergodic Roth theorem and applications

Durcik

Greenfeld

Iseli

et al. 2022

DCDS

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<p style='text-indent:20px;'>We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for <inline-formula><tex-math id="M1">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-systems for uniformly amenable groups <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>. As a special case, we obtain this uniformity over all <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-systems, and our result seems to be novel already in this case. Our uncountable Roth theorem is crucial in the proof of both of these results.</p>

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Section: Lemma B2 For Everymentioning

confidence: 99%

“…The converse implication does not hold. The group S ∞ of permutations of N which move only a finite number of elements is an example of a group which is amenable but not uniformly amenable, see [16]. See [43] for another example.…”

Section: Lemma B2 For Everymentioning

confidence: 99%